#
Section 5-1

#
Solving Linear Systems by Graphing

A
system of linear equations
is just a set of two or more linear equations.

In two variables, the graph of a system of two equations is a pair of lines in the plane. These lines will intersect in either zero points (if they are parallel), one point (in most cases), or infinitely many points (if the graph of the two equations is the same line).

No solutions:

*
y
*
=
**
–
**
2
*
x
*
+ 4

**
***
y
*
= –2
*
x
*
– 3

One solution:

**
***
y
*
= 0.5
*
x
*
+ 2

**
***
y =
*
–2
*
x
*
– 3

Infinitely many solutions:

**
***
y
*
= –2
*
x
*
– 4

**
***
y
*
+ 4 = –2
*
x
*

The
**
solution
**
of a system is the point of intersection, if it exists.

One easy way to find the solution of a system of equations is simply to graph the two lines, and then read the coordinates of the point of intersection.

For example, in the second graph shown above, the intersection point is (
**
–
**
2, 1).

**
One drawback of this method
**
is that you can't be sure you have the coordinates exactly. You can't be really sure that the intersection point is (
**
–
**
2, 1) and not (
**
–
**
2.001, 0.999). To be sure, you should check the point algebraically in both equations.

Substituting
*
x
*
=
**
–
**
2,
*
y
*
= 1 in
*
y
*
= 0.5
*
x
*
+ 2, we get:

**
1 = 0.5(–2) + 2
**

**
1 = –1 + 2
**

**
1 = 1
**

Substituting
*
x
*
=
**
–
**
2,
*
y
*
= 1 in
*
y
*
=
**
–
**
2
*
x
*
**
–
**
3, we get:

**
1 = –2(–2) – 3
**

**
1 = 4 – 3
**

**
1 = 1
**

So, in this case, the solution checks out.